The first and third rows of Figure REF  show that the four methods lead to comparable estimation and uncertainty quantification when estimating \(f_0\) . However, the deviation between different methods is considerably widened for the estimation of \(f_0^{\prime }\) . Matérn and squared exponential kernels constantly give the most accurate point estimation, and their credible bands cover the ground truth with reasonable width, indicating the effectiveness of the nonparametric plug-in procedure using GP priors. There is a tendency for the second-order Sobolev kernel to fail to capture \(f_0^{\prime }\)  around the left endpoint, particularly in Simulation 1. The performance of the B-spline method continues to exhibit sensitivity to the choice of \(N\) , selected by leave-one-out cross validation. In Simulation 1, the selected \(N\)  is 1, and the B-spline prior yields comparable credible bands of \(f_0^{\prime }\)  than GP priors with slightly altered estimation near zero; in Simulation 2 shown in the fourth row, the B-spline method with \(N=5\)  gives a point estimate that is substantially worse than the other three GP methods, and the associated credible bands are off the chart. We remark that the performance of B-splines might be substantially improved had the number of knots been selected by a different tuning method or with a different simulation setting. While leave-one-out cross validation may be appropriate for estimating \(f_0\) , as observed in [1]}, our results suggest that adjustments or alternative strategies seem to be needed when the objective is to make inference on \(f_0^{\prime }\) . For GP priors, our numerical results suggest choosing \(\lambda \)  using empirical Bayes seems to be a reasonable strategy for both \(f_0\)  and \(f_0^{\prime }\) , which is in line with the nonparametric plug-in property of GP priors.
